Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Stochastic Radon operators in porous media hydrodynamics


Authors: George Christakos and Dionissios T. Hristopulos
Journal: Quart. Appl. Math. 55 (1997), 89-112
MSC: Primary 76S05; Secondary 60G60, 76M35, 86A05
DOI: https://doi.org/10.1090/qam/1433754
MathSciNet review: MR1433754
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Abstract: A space transformation approach is established to study partial differential equations with space-dependent coefficients modelling porous media hydrodynamics. The approach reduces the original multi-dimensional problem to the one-dimensional space and is developed on the basis of Radon and Hilbert operators and generalized functions. In particular, the approach involves a generalized spectral decomposition that allows the derivation of space transformations of random field products. A Plancherel representation highlights the fact that the space transformation of the product of random fields inherently contains integration over a ``dummy'' hyperplane. Space transformation is first examined by means of a test problem, where the results are compared with the exact solutions obtained by a standard partial differential equation method. Then, exact solutions for the flow head potential in a heterogeneous porous medium are derived. The stochastic partial differential equation describing three-dimensional porous media hydrodynamics is reduced into a one-dimensional integro-differential equation involving the generalized space transformation of the head potential. Under certain conditions the latter can be further simplified to yield a first-order ordinary differential equation. Space transformation solutions for the head potential are compared with local solutions in the neighborhood of an expansion point which are derived by using finite-order Taylor series expansions of the hydraulic log-conductivity.


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DOI: https://doi.org/10.1090/qam/1433754
Article copyright: © Copyright 1997 American Mathematical Society


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